Stripline electromagnetic resonators consist of a center conductor sandwiched between two dielectric slabs. Outer surfaces of the dielectric slabs are in contact with ground planes which are conventionally made of metal. The center conductor, which is also conventionally a metal, as a length chosen to correspond with a fraction (approximately 1/2, 1/4, or 1/8) of the wavelength of the desired resonant frequency in the dielectric elements. Signals are coupled to and from the resonator using coupling mechanisms located laterally from the center conductor.
Recently, high-temperature superconducting materials have been used in electromagnetic resonators because of their low electrical surface resistance when cooled to below their critical temperatures. In the case of stripline resonators, the focus has been on the use of so-called thin film, high-temperature superconductors as both a material for the center conductor and the ground planes. Thin films are generally epitaxial, in which a single crystal of the high-temperature superconducting material is grown on a substrate. Thin film superconductors may have a thickness of about one micron but are usually only about 0.5 micron thick, after which they loose their epitaxy, and hence their desirable electromagnetic properties. Mannhart, J. et al., "High-T.sub.C Films, Growth Modes--Structure--Applications," NATO ASI Course on "Materials and Crystallographic Aspects of High T.sub.c Superconductivity" (1993 preprint).
The substrate in a thin film stripline resonator is chosen for its crystalline structure and serves as a template for the formation of the superconducting thin film. The crystalline structure of the substrate can be a limiting factor in the design of stripline filters because the substrate usually also serves as one of the dielectric slabs. A dielectric that has a suitable crystalline structure may not have a sufficiently high dielectric constant, or may have too high a dielectric loss, to be suitable as a dielectric element in a stripline filter. In addition, substrates must be chosen to minimize any chemical reaction between the superconductor and the substrate so that no undesirable reaction layer is formed between the superconductor and the substrate.
Another disadvantage of thin film superconductors is their relatively low ratio of the thickness of the film to electromagnetic penetration depth. The penetration depth is the depth below the surface of a superconductor at which an electromagnetic field external to the superconductor has been decreased by a factor of e (approximately 0.37). Penetration depth is temperature-dependent, with the smallest penetration depth for a material at 0 K. Penetration depth of superconductors can be determined for various temperatures using the formula .lambda..sub.T =.lambda..sub.0 /(1-(T/T.sub.c).sup.4).sup.1/2, where T is the temperature in Kelvin, .lambda..sub.T is the penetration depth at T, .lambda..sub.0 is the penetration depth at 0 K, and T.sub.c is the critical temperature of the superconductor. Shen, Z. Y., High Temperature Superconducting Microwave Circuits, Section 2.4.2, p. 29 (Artech House 1994). As used herein, penetration depth is measured at 77 K, which is the temperature at which many high-temperature superconductor devices are expected to operate. See, Apte, P. R. et al., "Microwave Surface Resistance of High T.sub.c Superconducting Films," High-Temperature Microwave Superconductors and Applications, Proc. SPIE, Vol. 2559, pp. 92-104, (Jul. 10, 1995). The strength of the field in a superconductor decreases exponentially so that some small amount of the field will penetrate through a thin film of superconducting material. With increased field penetration, the nonlinear power response increases, which, in turn, leads to increased distortion at higher powers or field strengths. See, Shen, Z. Y., High Temperature Superconducting Microwave Circuits, Section 2.8, pp. 47-57 (Artech House 1994). For instance, nonlinear responses at higher powers include excessive losses at the resonant frequency and increased intermodulation distortion (where signals at unwanted frequencies are produced from the interaction between two or more input signals).
Thin film superconductors exhibit a generally small penetration depth, as little as about 0.25 micron at 77 K. See Oates, D. E., "Surface Impedance Measurements of YBa.sub.2 Cu.sub.3 O.sub.7-x Thin Films in Stripline Resonators" IEEE Transactions on Magnetics, Vol. 27, pp. 867-871 (1991) (finding penetration depths of 0.167 micron at 4.2 K, which leads to a 0.275 micron penetration depth at 77 K). The small thickness of such films means that the ratio of thickness to penetration depth is at most 4:1 (when the film is one micron thick), but will usually be less than about 2:1 (when the film is less than 0.5 micron thick). In the case of a stripline resonator, the center conductor is subjected to magnetic fields both from its top and from its bottom so that its effective thickness for the purpose of comparison with penetration depth is only half the overall thickness of the thin film. With the thickness of the film effectively halved, the ratio of thickness to penetration depth for thin film stripline center conductors is only approximately 2:1 to 1:1. Thin film superconductors often exhibit a nonlinear power response (see Oates, D. E. et al., "Measurements and Modeling of Linear and Nonlinear Effects in Striplines", Journal of Superconductivity, Volume 5, pp. 361-369, August 1992), which may be caused by such a small thickness to penetration depth ratio.